We have studied the estimation process of quantum states from the measurement outcomes,and operationally defined the error and disturbance in quantum measurement by the accuracy of the estimation. We have shown several uncertainty relations between the measurement errors of two observables, and the uncertainty relations between the error and disturbance.

In 1927, W. Heisenberg discussed a thought experiment about the position measurement of a particle by using a γ-ray microscope. He argued that more accurately the position is measured, the more strongly the momentum of the particle is disturbed by the backaction of the measurement by using semi-classical analysis, and derived a trade-off relation between the error ε(x) in the measured position x and the disturbance η(px) in the momentum px caused by the measurement process:

This inequality epitomizes the complementarity in quantum measurements: we cannot perform a measurement of an observable without causing disturbance in its canonically conjugate observable. However, this analysis was not fully quantum mechanical.

In the early days of quantum mechanics, the Kennard-Robertson inequality

was erroneously interpreted as a mathematical formulation of the trade-off relation between the error and disturbance in quantum measurement, where <A> := Tr[pA] is the expectation value of A over the quantum state p, the square bracket denotes the commutator [ A, B ] := AB - B A, and σ(A)2 := <A2> - <A>2 . The Kennard-Robertson inequality actually implies the indeterminacy of the quantum state: non-commuting observables cannot have definite values simultaneously. However, since σ(A) does not depend on the measurement process, the Kennard-Robertson inequality re ects the inherent nature of a quantum state alone, and does not concern any trade-off relation between the error and disturbance in the measurement process.

In 1988, E. Arthurs and M. S. Goodman considered a simultaneous measurement of two non-commuting observables A and B in a fully quantum mechanical treatment. Because A and B do not commute with each other, it is necessary to extend the Hilbert space to make both of them simultaneously measurable. This can be done by letting the system interact with another system, called the apparatus. By considering the interaction between the system and apparatus, they considered an indirect measurement. In order to make the outcomes of the indirect measurement meaningful for A and B , they assumed the unbiasedness of the measurement outcomes. That is, the expectation values of the outcomes respectively equal to < A> and <B > for an arbitrary quantum state. The unbiasedness of the measurement implies that <A> can be estimated directly from the distribution of the measurement outcomes. They derived that the variances of the measurement outcomes satisfy

Comparing this result with the Kennard-Robertson inequality, we find that the lower bound is doubled. In the model of Arthurs nad Goodman, the sources of the fluctuations of the measure- ment outcomes are both the system's inherent fluctuations and the error in the measurement process. Each source of the fluctuations has the lower bound of 1/ 2 |<[ A, B]>| , and the bound in (3) is doubled. Because the measurement discussed by Arthurs and Goodman is restricted to the unbiased measurement, a natural question arises as to what happens for the biased measurement case.

M. Ozawa generalized the Arthurs-Goodman inequality by removing the unbiasedness condition, and derived the following inequality:

Because the error ε(A) is always finite, if the error ε(A) vanishes, the product of the measurement errors also vanishes. Thus, the Heisenberg-type trade-off relation can be violated:

However, his definition of the measurement error does not correspond to the accuracy of the estimation any more due to the removal of the unbiasedness condition. For example, if the outcome of a measurement is fixed for an arbitrary state p, the error ε(A) can vanish even if we cannot estimate <A>. Such a result originates from ignoring the estimation process which must inevitably be accompanied in the unbiased measurement. Ozawa also defined the disturbance η(B ) caused by the backaction of the measurement, and derived the following inequality:

However, his definition of the disturbance does not involve the estimation process either. We consider that to analyze the error and disturbance in quantum measurement, it is crucial to clarify the role of the estimation which must be made on the measurement outcomes.

Estimation theory provide us a description of how accurately we can estimate values and how much information we can obtain from realizations of the probabilistic phenomena. In quantum theory, measurements on the quantum system are necessary to obtain some pieces of information about the quantum system, and the measurement outcomes are obtained according to the probability distribution. Thus, it is necessary to involve the estimation theory for clarifying the uncertainty relations about the error and disturbance in quantum measurement. In estimation theory, one of the most important quantities is the Fisher information, which gives the upper bound on the accuracy of the estimated value. The probability distribution of the outcomes is determined by the measured quantum state and the choise of the measurement operators. The probability distribution and the Fisher information vary with varying the measurement. The quantum Fisher information gives the upper bound of the Fisher information obtained by the measurement.

We develop a general theory of the error and disturbance in quantum measurement from the viewpoint of quantum estimation theory. We show that the unbiasedness is not necessary for the measurements, but for the estimation from the measurement outcomes. We analyze that the estimated values from measurement outcomes consist of three types of errors: inherent uctuation of an observable, error in the measurement, and the estimation error caused by the unoptimality of the estimator. We extract the measurement error from the variance of the estimator in terms of the Fisher information, which gives the upper bound of the accuracy of the estimator. Our definition of the measurement error reduces to Arthurs-Goodman's definition for the case of the unbiased measurement. We also analyze that disturbance caused by the backaction of the measurement is quantified in terms of the loss of the Fisher information contents during the measurement process.

By using our definition of the error and disturbance, we prove that the following uncertainty relation between the errors of two observables

and the uncertainty relation between the error and disturbance

However, the lower bounds, in general, cannot be attained in these inequalities. By introducing quantities σQ(A) and CQS( A, B ), which respectively characterize the quantum fluctuations of A and the quantum correlation between A and B , we derive new inequalities providing the attainable bounds:

If we focus on the observable A and B , they are inherently uctuated in general. Such fluctuations can be separated to the two parts: classical fluctuations and quantum fluctuations. The value σQ(A) characterizes the uctuation which originate from the quantum mechanics. The quantum correlation CQS( A, B ) also characterizes the correlation of the observable which originate form the quantum mechanics. We have proved that these bounds can be attained for the 2-dimensional Hilbert space, and for a class of measurements for higher dimensions. We have shown numerical evidences for the validity of the bounds for an arbitrary measurement for the higher-dimensional Hilbert spaces.

本論文は、全10章からなる。第1章は不確定性関係に関する導入、第2章は不確定性関係に関する先行研究の概説、第3章と第4章がそれぞれ古典推定理論と量子推定理論の概説、第5章がLie代数の生成子を用いた離散量子系の記述、第6章がLie代数を用いたFisher情報量および量子Fisher情報量の解析、第7章が任意の量子測定における測定誤差と擾乱の定式化、第8章が測定誤差と2つの物理量の測定誤差に対する不確定性関係、第9章が量子測定における誤差と擾乱の間の不確定性関係、そして第10章が結論と討論という構成である。

1927年に提案されたHeisenbergの不確定性関係は、γ線顕微鏡を用いて粒子の位置を測定する思考実験から導かれた、位置の測定誤差と運動量の擾乱との積はPlanck定数で決まる一定の値よりも小さくすることができない、ということを表す不等式である。この不確定性関係は、ある物理量を測定するとその反作用により正準共役な物理量に擾乱を与えるという、量子力学における測定の相補性を表すと解釈され、量子力学と古典力学との本質的な相違を記述するものと考えられてきた。しかしこの不等式は、半古典論に基づいて導出されたものであり、完全に量子力学的な記述を行なって導出したものではなかった。一方、2つの物理量の分散の積を物理量の交換関係で決まる値より小さくすることはできない、というKennard-Robinsonの不等式を不確定性関係と呼ぶ場合もあるが、この関係式は測定誤差に関すると擾乱の間の不確定性関係を記述するものではない。量子測定理論に基づいた測定誤差に対する不確定性関係は、1988年にArthursとGoodmanによって不偏測定という特別なタイプの量子測定を用いた間接測定モデルに対して定式化され、測定誤差の効果により2つの物理量の分散の積はKennard-Robinson型の不等式の2倍の下限を持つというHeisenberg型の不確定性関係が示された。これに対して、2004年に小澤は、ArthursとGoodmanのモデルを任意の間接測定に拡張し、測定誤差を記述する関数と擾乱を記述する関数を定義することによって、測定誤差と物理量の分散、そして擾乱の関係を示す不等式を導出した。この不等式は、Heisenberg型の不確定性関係を破る場合がある。ArthursとGoodmanが定義した測定誤差を記述する関数は、測定によって得られる量子系の情報量に対する誤差という操作論的な意味を持っていたが、小澤の定式化では任意の間接測定への拡張が行なわれたことにより、測定誤差と擾乱を記述する関数の操作論的な意味が不明確となっていた。

本論文は、測定によって量子系の情報をどのぐらい知ることができるかという視点から、量子推定理論に基づき量子測定における測定誤差と擾乱をFisher情報量および量子Fisher情報量を用いて定義し、これらの間に成り立つ不確定性関係を導いたものである。第6章で導出された離散量子系におけるLie代数の生成子を用いたFisher情報量および量子Fisher情報量の記述により、量子推定理論の立場から第7章で定義された任意の量子測定における測定誤差と擾乱の関数が解析を容易にする簡潔な形で導かれている。そして、第8章において、2つの物理量の測定誤差に関するHeisenberg型の不確定性関係を導き、さらに、2つの物理量の間の量子相関を記述する関数を導入することで、達成可能な下限を示す新たな不等式が導出された。第9章では、測定誤差と擾乱に関するHeisenberg型の不確定性関係が導かれ、さらに、2つの物理量の間の量子相関を記述する関数を用いて、測定誤差と擾乱に関する達成可能な下限を示す新たな不等式が導出されている。また、どのような測定を行なっても、量子2準位系においては、第8章と第9章で得られた達成可能な下限を示す新しい不等式が成立することを解析的に証明し、第8章で導いた新しい不等式については、量子3準位系から量子7準位系までの量子系で成立することを数値的に示した。

このように、情報論的な観点から不確定性を掘り下げることによって、一般的な量子測定に対して操作論的な意味を持つ測定誤差と擾乱を定式化し、その不確定性関係を導いた本論文の結果は、量子力学の本質的な理解に大きな知見を与える画期的な成果であると考える。

なお、本論文第6章から第9章までの研究は、上田正仁教授との共同研究、第7章と第8章の一部は、上田正仁教授および沙川貴大との共同研究であるが、論文提出者が主体となって分析および検証を行ったもので、論文提出者の寄与が十分であると判断する。

したがって、博士(理学)の学位を授与できると認める。